Nonsmooth dynamics in spiking neuron models

Nonsmooth dynamics in spiking neuron models

S Coombes, R Thul, K C A Wedgwood

School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK.

Abstract

Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysio- logical properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a nonsmooth dynamical system. In this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate (ranging from tonic to burst firing). We show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis, particularly those for treating grazing bifurcations. Importantly we highlight one particular single neuron model, capable of generating realistic spike trains, that is both computationally cheap and analytically tractable. This is a planar nonlinear integrate-and-fire model with a piecewise linear vector field and a state dependent reset upon spiking. We call this the PWL-IF model and analyse it at both the single neuron and network level. The techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model, as well as to develop the notion of Liapunov exponents. We also show how to construct the phase response curve for this system, emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems. The stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times. At the network level we consider linear coupling between voltage variables, as would occur in neurobiological networks with gap-junction coupling, and show how to analyse the properties (existence and stability) of both the asynchronous and synchronous states. In the former case we use a phase-density technique that is valid for any large system of globally coupled limit cycle oscillators, whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking. Finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics.

Key words: integrate-and-fire, spiking neuron model, nonsmooth bifurcation, linear-coupling.

1. Introduction

Spiking neurons are at the heart of many computa- tional models of the brain that aim to improve our un- derstanding of brain function and dysfunction. The Blue Brain Project [1] is a case in point. This has utilised IBM’s Blue Gene parallel supercomputer to attempt the construction of a biologically accurate model of neural tis- sue from first principles. At present initial simulations of

∼ 10⁴biophysically detailed neurons have been performed, setting the scale of the tissue at roughly one neocortical column. Given that a whole human brain contains 10¹⁰ neurons there has been a push in the computational neuro- science community to develop complimentary models that are reduced in their complexity yet still able to generate

Email addresses: [email protected] (S Coombes), [email protected] (R Thul), [email protected] (K C A Wedgwood)

the rich repertoire of behaviour seen in a real nervous sys- tem. Perhaps the most famous example of such a model is the FitzHugh–Nagumo model [2], comprising two cou- pled ordinary differential equations for the generation of continuous action potential like shapes of spiking voltage activity. In this case analytical progress has also been possible with one further step, namely, the introduction of piecewise linear (PWL) nullclines. This gives rise to the so-called McKean model [3], for which a number of results about the existence and stability of periodic orbits are now known [4, 5, 6]. Indeed there are now a number of planar PWL single neuron models for mimicking the behaviour of tonically firing neurons, and we refer the reader to [7]

for a recent discussion. Moreover, the PWL nature of such models means that techniques from nonsmooth dynamics are particularly relevant to their analysis, and indeed re- cent progress on understanding canard explosions has been made by studying PWL models of FitzHugh-Nagumo type [8]. However, the spiking patterns of such planar models

are typically not as diverse as one needs to mimic realistic firing patterns, such as bursting.

The currently most successful class of minimal models that satisfy the criterion of being able to generate realis- tic firing patterns are those of integrate-and-fire (IF) type, where a simple threshold unit is used to caricature the excitable aspect of real cells that gives rise to an action potential spike. In these models the spike shape is discon- tinuous. Recent work by Izhikevich has developed a large- scale thalamo-cortical model with ∼ 10⁶ neurons using a phenomenological two dimensional nonlinear IF model [9].

One key aspect of any IF model is the discontinuous reset of a state variable upon reaching some threshold for spik- ing. It is this particular harsh nonlinearity in the dynamics that endows these models with interesting dynamics and precludes their description using the machinery of smooth dynamical systems. Indeed they have much in common with models of impacting systems that have been devel- oped for the study of mechanical structures such as rocking blocks [10], rattling gear boxes [11] and print hammers [12].

For a discussion of impacting systems in general we refer the reader to the recent book by di Bernardo et al. [13].

Thus it is timely to revisit the dynamics of IF models us- ing the techniques developed for the study of more general nonsmooth systems, such as those reviewed in [14], and develop the mathematical insight into network behaviour that can complement simulation studies that are being per- formed in the computational neuroscience community.

In section 2 we provide a review of some of the more popular models of IF type that are currently being used as models of spiking neurons. To illustrate that nonsmooth bifurcations play a fundamental role in the description of their behaviour we present an analysis of the periodically forced leaky IF model in section 3. Here we show that grazing bifurcations are especially important in determin- ing the Arnol’d tongue diagram for mode-locked responses, and note the relevance of this to modelling spike trains in the sensory periphery. For modelling the spike trains in deeper brain regions, such as the cortex, we introduce a new class of IF model in section 4. This model is able to reproduce a range of spiking patterns, from tonic to burst firing, yet is analytically tractable. In essence the model below the threshold for firing evolves according to a planar PWL dynamical system. We present an original bifurca- tion analysis of this model in response to constant current injection focusing on local discontinuity induced bifurca- tions. Next in section 5 we show how to construct periodic orbits and determine their stability as well as calculate the phase response curve (by adapting techniques originally developed for the analysis of limit cycles in smooth dy- namical systems). In section 6 spike-adding bifurcations (for bursting orbits) are described in terms of bifurcations of an associated one-dimensional return map. The no- tion of Liapunov exponents is developed in section 7, using techniques originally developed for the analysis of impact oscillators. Next in section 8 we turn to the construction and analysis of neural networks. We focus on gap-junction

coupling, where the natural way to describe electrically in- teracting cells is via an ohmic resistance, which translates into a linear coupling between voltage state variables. For large globally coupled networks we show how to determine the properties of the synchronous and asynchronous states (existence and stability). Finally we end with a discussion of future challenges in the understanding of neurodynam- ical systems that are likely to benefit from further cross- over of ideas from nonsmooth dynamical systems.

2. A review of integrate-and-fire models

Although conductance-based models like that of Hodgkin and Huxley [15] provide a level of detail that helps us to understand how neural cells generate action-potential elec- trical spikes their high dimensionality (four for Hodgkin- Huxley though rising to hundreds for compartmental mod- els that express realistic ionic currents) precludes them from detailed study, especially at the network level. Thus simpler models are more appealing, especially if they can be fit to single neuron data. It is now known that nonlin- ear extensions of the basic leaky IF model can accurately fit intracellular voltage recordings [16]. A one-dimensional nonlinear IF model takes the form

dv

dt = f (v) + I(t), (1)

such that v is reset to vRjust after reaching the threshold value vth> vR. Here v is interpreted as a voltage variable and I(t) is an external drive (that might be under the control of an experimentalist or arise from the activity of other neurons to which a cell is coupled). Firing times are defined iteratively according to

T_n= inf{t |v(t) ≥ v_th ; t ≥ T_n−1}. (2) One-dimensional IF models with a fixed voltage threshold are caricatures of excitable neural systems and as such it is well to mention that they cannot adequately capture the refractory properties of real neurons. This is often achieved with the introduction of an absolute time during which they cannot fire after reaching threshold or by the introduction of a time dependent threshold that increases after a firing event and makes it harder for the neuron to subsequently fire (mimicking a relative refractory pe- riod), as reviewed in [17]. Moreover, real neurons (and Hodgkin-Huxley style models) do not possess a fixed volt- age threshold, and firing ultimately depends on the state of receptors within a membrane. Although differential equa- tions for the threshold in IF models can be found that mimic more closely the properties of real neurons [18], we limit our discussion in this paper to models with a constant threshold.

2.1. Leaky IF model

The leaky IF model (LIF) is attributed to Lapicque in 1907, although the phrase “integrate-and-fire” was first 2

coined by Bruce Knight in the 1960s [19]. It is defined by (1) and (2) with the choice

f (v) = −v

τ, τ > 0. (3)

Because of its linear nature we may solve the sub-threshold dynamics of the model exactly for v < vthwith initial data v(t0) < vth at time t = t0 (using an integrating factor, variation of constants or Green’s function):

v(t) = v(t₀)e^{−(t−t0)/τ}+ Z t

t0

e^{−(t−s)/τ}I(s)ds. (4) For a periodic input the system may well respond peri- odically though without reaching threshold. This is com- monly referred to as a sub-threshold oscillation and is to be distinguished from the case when oscillations arise via the reset mechanism. Consider in particular the case of a constant drive, where the threshold can only be reached from v(t0) if Iτ > vth. The threshold will subsequently be reached from vR and a periodic oscillation will occur.

The period of oscillation ∆ = Tn+1− Tn is determined by setting v(T_n+1) = v_th with v(T_n) = v_R, giving

∆ = τ ln Iτ − v_R Iτ − vth



H(Iτ − v_th), (5) where H is the Heaviside step function. The inclusion of the Heaviside term reflects the fact that oscillations do not occur for Iτ < v_th. Electrophysiologists often classify neu- ron response in terms of the so-called f − I curve, which shows the frequency of oscillation as a function of the time independent drive I. For the LIF model this is easily con- structed from (5) using f = ∆⁻¹, showing a sharp rise in f (from zero) as I increase through the critical value vth/τ . A plot of the response of the LIF model to con- stant drive I is shown in Fig. 1. Here one sees that the model does not capture the essential shape of a real action potential. Rather the IF model is deemed to be good at capturing the time of generation of an action potential.

Since many models of synaptic (chemical) interaction are based on spike-times, rather than spike shapes, this favours the IF model in large scale simulations of synaptically cou- pled neurons. The tractability of this single neuron model (linear dynamics between firing events) means that it is particularly suited to analysis at the network level with event based models of chemical synapses. Indeed a theory of phase-locked behaviour for strong coupling has been de- veloped for just this scenario [20]. However, gap junction (linear) coupling between neurons means that the action potential shape is communicated from one neuron to an- other and so LIF models are (without modification) poor candidates for use in this case.

2.2. Nonlinear IF models

The quadratic IF (QIF) neuron is the simplest gener- alisation of the LIF model that captures qualitatively the

0 1

0 1 2 3 4 5

v

t

Figure 1: Voltage trace for an LIF oscillator with constant drive I = 2 with τ = 1, vth= 1 and vR= 0.

0 5 10

0 3 6 9 12 15

v

t

Figure 2: Voltage trace for the QIF oscillator with constant drive I = 1 with vth= 10 and vR= −1.

behaviour of the f − I curve of a large family of more real- istic models [21]. Interestingly, this model was apparently already known to Alan Hodgkin, and used to fit some of his data (and also subsequently analysed by Bruce Knight).

Up to shifts and constant factors it is defined by

f (v) = v². (6)

Unlike the LIF model the QIF does allow a representation of an action potential shape (for I > 0 the voltage rises sharply to threshold), as shown in Fig. 2. For I < 0 there are two equilibria (one stable and the other unstable) and for I > 0 these disappear via a saddle-node bifurcation at I = 0. In the oscillatory regime (I > 0) the trajectory (for constant drive) can be integrated for T_n < t < T_n+1 to give

v(t) =√ I tan



tan⁻¹ v_R

√ I

 +√

I(t − T_n)

 . (7) The period of oscillation is calculated by setting v(T_n+1) = vth with v(Tn) = vRgiving

∆ = 1

√I



tan⁻¹ vth

√I



− tan⁻¹ vR

√I



H(I).

In the limit v_th→ ∞ and v_R→ −∞ we see that ∆ = π/√ I (and we have blowup of the voltage trajectory in finite time), and the f − I curve shows a√

I dependence, which matches many cortical neurons much better than the LIF f − I curve. For a further discussion of this model we refer the reader to the book by Izhikevich [22].

With the improvement in neuronal modelling by simply changing the shape of the nonlinearity from (3) to (6) this raises the question as to whether more judicious choices can improve things further still. Interestingly Fourcaud- Trocm´e et al. [23] have shown that choosing f (v) = exp(v)

-80 -40 40

19.5 20.5 21.5 22.5

mV

t

t

Figure 3: Sample voltage traces (mV) as a function of time (seconds) from the linear-exponential IF model (green dashed line) and data (red solid line) from a layer-5 pyramidal cell in response to a noisy current injection (constructed from two summed Ornstein-Uhlenbeck processes, see [16] for further details).

(up to shifts and scaling) can act as an approximation of a more detailed conductance-based spiking model. In fact it has now been shown that real cortical data (from layer-5 pyramidal cells) can be very accurately fit with the following choice [16]:

f (v) = −1

τ(v − vL) +κ

τe^(v−vκ)/κ, (8) with vth = 30.0, vR = −71.2, vL = −68.5, τ = 3.3, vκ= −61.5 and κ = 4. Fig. 3 nicely illustrates the strong fit of the model to real data for a stimulation protocol which is a noisy current injection. Similarly to the QIF model the linear-exponential IF (LEIF) model obtained using (8) has two equilibria (defined by f (v)+I = 0) which disappear in a saddle-node bifurcation when I = −f (v^∗), where v^∗ is defined by f⁰(v^∗) = 0. In common with the QIF model it is able to support oscillations with arbitrar- ily low frequency just beyond the bifurcation point. Both the QIF and LEIF models have only a weak dependence on the choice of threshold value since they both blow up in finite time (in the absence of a threshold).

2.3. Planar IF models

Unfortunately, one dimensional nonlinear IF models, as they stand, are unable to reproduce bursting patterns of activity, which are typically associated with slow calcium dependent processes. One way to incorporate such a slow process is by coupling the voltage dynamics to a recovery or adaptive process in the following manner:

dv

dt = f (v) − a + I, 1 ω

da

dt = βv − a. (9) Here the parameters β and ω, respectively, describe the sensitivity and decay rate of the adaptive process. Upon reaching threshold the voltage is reset (v → vR) and a is adjusted according to a → a + k. The Izhikevich model [24, 25] is one such model with f (v) = 0.04v²+ 5v + 140.

Interestingly this model can capture a number of neu- ronal firing patterns including tonic (repetitive) spiking, bursting and fast spiking as illustrated in Fig. 4, despite its sensitivity to the choice of threshold value [26]. It is

-60 -40 -20 0 20 40 V

0 50 100 150 200

t i)

-60 -40 -20 0 20 40 V

0 50 100 150 200

t ii)

-60 -40 -20 0 20 40 V

0 50 100 150 200

t iii)

-60 -40 -20 0 20 40 V

0 50 100 150 200

t iv)

Figure 4: Firing patterns in the Izhikevich model with I = 10 and vth = 30. Voltage traces as a function of time for i) tonic spiking (α = 0.02, β = 0.2, vR= −65, k = 8), ii) tonic spiking (α = 0.02, β = 0.2, vR = −55, k = 4), iii) bursting (α = 0.02, β = 0.2, vR= −50, k = 2), and iv) fast spiking (α = 0.1, β = 0.2, vR= −65, k = 2).

worth noting that a similar model to that of Izhikevich was independently introduced by Gr¨obler et al. [27] as a model of a pyramidal cell in hippocampus CA3. The adap- tive exponential integrate-and-fire model is obtained us- ing a linear-exponential term for f (v) (as in equation (8)) [28, 29], whilst the quartic model is obtained by choosing f (v) = v⁴+ 2ωv [30]. Both are able to produce a wide va- riety of firing patterns, and the quartic model in particular has a very nice repertoire of responses ranging from tonic spiking to bursting as well supporting phasic responses, rebound, spike frequency adaptation, sub-threshold oscil- lations and much more, all of which are discussed in detail in [30].

Apart from the LIF model none of the models described above admits to closed form solutions for arbitrary (non- constant) drive. A somewhat overlooked tractable (one dimensional) nonlinear IF model is that of Karbowski and Kopell [31], with a nonlinearity given by f (v) = |v|, which we shall call the absolute IF model (AIF). Because of the choice of a PWL form of the nonlinearity the AIF model can be explicitly analysed. Moreover, it is also capable of generating behaviour consistent with that of a fast-spiking interneuron [32]. The generalisation of the model to allow for bursting behaviour is easily achieved by extending it to the form of (9). A minimal AIF model with adapta- tion is obtained for f (v) = |v| and β = 0. For sufficiently small k the model fires tonically and for larger values of k the model can also fire in a burst mode. The mecha- nism for this behavior in the AIF model (and indeed all planar models discussed here) is most easily understood in reference to the geometry of the phase-plane. We il- lustrate, in Fig. 5, the phase plane for the AIF model,

4

0 1

0 10 20 30 40 50

v

t 0 1

0 100 200 300 400 500

v

t

0 0.1 0.2 0.3 0.4

0 0.4 0.8 v

a

0 0.1 0.2 0.3 0.4

0 0.4 0.8 v

a

Figure 5: Top left: Tonic firing in the AIF model with spike adap- tation. Here ω = 1/3 and k = 0.75ω. Top right: Burst firing in the AIF model with spike adaptation. Here ω = 1/75 and k = 2ω.

Bottom left: A periodic orbit in the (v, a) plane corresponding to the tonic spiking trajectory shown above (green curve). Also shown is the voltage nullcline (red lines) as well as the value of the reset (black dashed line). Bottom right: Burst firing in the AIF model with spike adaptation. Here ω = 1/75, and k = 2ω. Other parame- ters are vR= 0.2, vth= 1 and I = 0.1.

and refer the reader to [32] for a more detailed discussion and analysis of this model. The analysis of how parame- ter space partitions into tonic, 1-spike per burst, 2-spike per burst, etc. firing patterns is an open mathematical (classification) challenge. It is worth noting that all the planar models considered here have much in common and can generate a very similar repertoire of firing behaviours, though the AIF model does not have trajectories that blow up in finite time (in the absence of a threshold).

3. Bifurcations of the periodically forced LIF neu- ron

Because all IF models include a threshold process spikes can be created or annihilated as a voltage trajectory tan- gentially intersects the threshold. This is naturally the case when a time-varying current injection (such as a pe- riodically varying synaptic current) is considered (and not just a constant drive). Thus it becomes important not only to assess the stability of spike trains to perturbations that leave the number of spikes unchanged (though do mod- ify firing times), but to address any instabilities that may arise via nonsmooth grazing bifurcations. To show how this can be done we present an analysis of the periodically forced LIF model, though stress that the ideas we present here carry over to more complicated IF models such as those reviewed in section 2.

The phenomenon of mode-locking is well documented in the literature on the periodic forcing of nonlinear oscil- lators. It is most commonly studied in the context of the standard circle map (see for example [33]). This map is known to support regions of parameter space where the ro- tation number (average rotation per map iterate) takes the value p/q, where p, q ∈ Z⁺. These regions are referred to as p:q Arnol’d tongues. In a neural context mode-locked

solutions are simply identically recurring firing patterns for which a neuron fires p spikes for every q cycles of forc- ing. With an increase of the coupling amplitude from zero Arnol’d tongues in the standard circle map typically open as a wedge, centered at points in parameter space where the natural frequency of the oscillator is rational. In be- tween tongues quasi-periodic behaviour, emanating from irrational points on the amplitude/frequency axis, are ob- served. The technique for calculating such tongues in IF models was first developed by Keener et al. [34] and later expanded upon in [35, 36].

Consider a LIF neuron with threshold at vth = 1 and reset level vR = 0 being driven by a ∆ periodic signal I(t) = I(t + ∆). An implicit map of the firing times may be obtained by integrating between reset and threshold according to equation (4). Introducing the function

G(t) = Z 0

−∞

e^s/τI(t + s)ds, G(t) = G(t + ∆), (10) gives

e^Tn+1/τ[G(T_n+1) − 1] = e^Tn/τG(T_n). (11) Defining

F (t) = e^t/τ[G(t) − 1], (12) we obtain an implicit map of the firing times in the form

F (Tn+1) = F (Tn) + e^Tn/τ. (13) A 1:1 mode-locked solution is defined by Tn = (n + φ)∆, with φ ∈ [0, 1), giving a fixed point equation

G(φ∆) = 1

1 − e^−∆/τ. (14)

Stability is examined by considering perturbations of the form Tn→ Tn+ δn, giving

δ_n+1= κ(φ)δ_n, κ(φ) = e^−∆/τ I(φ∆)

I(φ∆) − 1/τ. (15) Solutions are stable if |κ(φ)| < 1 The borders of the re- gions where 1:1 solutions become unstable are defined by κ(φ) = 1 (tangent bifurcation) and κ(φ) = −1 (period doubling bifurcation). However, solutions may also lose stability in a nonsmooth fashion in two possible ways, which we shall refer to as type (a) and type (b). In type (a) there is a tangential intersection of the trajectory with the threshold value such that upon variation of the bifurca- tion parameter the local maxima of the voltage trajectory passes through threshold from above. This is defined by

˙v = −v/τ + I = 0, so that I(Tn) = 1/τ or equivalently F⁰(Tn) = 0. In type (b) a sub-threshold local maxima in- creases through threshold leading to the creation of a new firing event at some earlier time than usual. This is defined by F (T^∗) = F (Tn)+e^Tn/τand F⁰(T^∗) = 0 with T^∗< Tn+1

and T_n+1is the solution to F (T_n+1) = F (T_n) + e^Tn/τ. As an example consider the choice

I(t) = I₀+

(+ 0 ≤ t < ∆/2

− ∆/2 < t < ∆. (16)

In this case the condition |κ(φ)| = 1 is independent of φ, since I(φ) = I₀± . A tangent bifurcation occurs when κ = 1:

± = −I0+ 1/τ

1 − e^−∆/τ. (17)

A nonsmooth bifurcation of type (b) is defined by the two

0 1 2

0 1 1.5 I 2

E

1:1

K K

Type (b)

$T

0 0 1 2 3

0.6 1 1.4 1.8

ε

τ

1:1 3:4 2:3 3:5

Figure 6: Left: 1:1 Arnol’d tongue in the LIF model with I(t) a

∆-periodic square wave with amplitude I0± . Note that a type (b) nonsmooth bifurcation significantly shapes the tongue structure.

Here τ = 1 and ∆ = 2. Right: p:q Arnol’d tongues in the LIF model with I(t) = I0+  sin 2πt. Here I0= 2. Below the dashed line the firing map is invertible.

equations

τ (I + )(1 − e−∆(1/2−φ)/τ) = 1 (18) ve^−φ∆/τ+ τ (I + )(1 − e^−φ∆/τ) = 1, (19) where 0 < φ < 1/2 and

v = e^−∆/2τ+ τ (I − )(1 − e^−∆/2τ). (20) Between them the above two bifurcations define the 1:1 Arnol’d tongue as shown in Fig. 6 (left) (period doubling and type (a) bifurcations are not possible for the param- eter values shown). The construction of other tongues with more general values of p:q is carried out in [35, 36], and the resultant tongue structure calculated for I(t) = I₀+  sin 2πt is shown in Fig. 6 (right). Once again the right hand borders of Arnol’d tongues are defined by type (b) nonsmooth bifurcations (and all others by tangent bi- furcations of the firing map).

In a pair of recent papers [37, 38] it has been shown that spike data from stellate cells in the ventral cochlear nu- cleus are very well explained by a LIF model with thresh- old noise, and that Arnol’d tongues are a practical way to understand the way in which single cells in the audi- tory periphery encode periodic stimuli. Indeed responses of LIF models to chaotic forcing have also been shown to be largely determined by grazing bifurcations [39]. The tech- niques described above have also been applied to several variants of the LIF model, including the IF-or-burst model [40], the “ghostburster” model [41] and the resonate-and- fire neuron model [42] as well as to PWL neuron mod- els [43]. Most recently an IF model with a slow T-type calcium current has been studied and been shown to sup- port chaotic behaviour in response to periodic forcing [44].

Interestingly by determining the condition for a grazing

bifurcation it was shown that knowledge of unstable pe- riodic orbits (existence and stability) could be combined with the grazing condition to determine an effective one- dimensional map that captured the essentials of the chaotic behavior. This map is discontinuous and has strong sim- ilarities with the universal limit mapping in grazing bi- furcations derived in the context of impacting mechanical systems [45]. This latter map was derived for grazing bi- furcations that occur in impacting mechanical oscillators and can support period adding cascades with or without chaotic bands.

4. A piecewise linear IF model

The aspect of the LIF model that allows one to perform an analysis such as the one above is obviously its linear- ity (below threshold). A similar analysis for say the QIF, LEIF or Izhikevich model would be much harder owing to the inherent nonlinear nature of these models. However, the AIF model described in section 2 is a natural starting point for the development of a more general PWL spiking neuron model that can be explicitly analysed. The use of PWL modelling is already quite common in neuroscience, with the McKean model [3] being a classic example. This may be regarded as a variant of the FitzHugh-Nagumo model [2] that provides a planar model of an excitable cell in which the dynamics is broken into simpler linear pieces.

An extension of this approach to develop PWL caricatures of other single neuron models, including the Morris-Lecar model, has recently been pursued by Tonnelier and Gerst- ner [46] and Coombes [7].

In this section we advocate a new type of PWL IF model, that we shall call the PWL-IF model. It is a gener- alisation of the AIF model with adaptation that we write in the form of (9) with

f (v) =

(v v ≥ 0

−sv v < 0. (21)

For a constant drive I the model may exhibit a number of different periodic attractors, and in particular we distin- guish between those that remain sub-threshold, and those that cross threshold, which we shall term spiking solutions.

We make further distinctions between spiking solutions as follows.

Fast spiking orbits: Attracting limit cycles which have v > 0 along the entire orbit and which have v(t^∗) = v_that precisely one value of t^∗ ∈ [t, t + ∆] ∀t, where ∆ is the period of the limit cycle.

Regular (or tonic) spiking orbits: Attracting limit cycles which have v < 0 for some segment of the or- bit and which have v(t^∗) = vth at precisely one value of t^∗∈ [t, t + ∆] ∀t, where ∆ is the period of the limit cycle.

n-Spike bursting orbits: Attracting limit cycles which have v < 0 for some segment of the orbit and which have v(t^∗) = vth at precisely n values of t^∗ ∈ [t, t + ∆] ∀t, where ∆ is the period of the limit cycle.

6

Sub-threshold oscilations: Attracting limit cycles which have v < v_th along the entire orbit.

The fast spiking orbits are so called as they may have arbitrarily fast frequency, whereas the frequency of regu- lar spiking orbits must be finite. With increasing I the model can make a transition from regular to fast spiking.

Contrary to the case for smooth systems, periodic orbits in discontinuous systems need not enclose a fixed point. In fact, the reset mechanism of the PWL-IF model allows for periodic orbits of (9) in the absence of any fixed points.

For β < 1, the f − I curve (regular spiking) reaches a maximum value before a bifurcation to fast spiking oc- curs. The switch between the two modes for β > 1 may have a further signature of doublet (2-spike burst) firing (which we shall consider in more detail below), and leads to a discontinuous f − I curve. Figure 7 depicts the f − I curve for differing values of β under variation of I. We can clearly see the transitions between the different oscillatory regimes, particularly for β = 1.2, where we observe discon- tinuities in the frequency response at a grazing bifurcation, and at the onset and termination of doublet firing.

0.04 0.08 0.12 0.16 0.2 0.24 0.28

0 2 4 6 8 10 12 14

I

f

Termination of doublet firing

0.04 0.08 0.12 0.16

0 0.5 1 1.5 2

I

f

Figure 7: Variation of the firing frequency under variation of the drive I for: Top: β = 1.2, Bottom: β = 0.9. We can clearly see how the firing rate changes as we move between solution types, and that the firing rate of the model during fast spiking is much more sensitive to changes in I than in the regular spiking mode.

In order to characterise where in parameter space dif- ferent types of solution exist, it is useful to consider the dif- ferent types of bifurcation that can occur. The v-nullcline has a characteristic ‘V’ shape, whilst the a-nullcline is a straight line with slope β. By inspection, we see that there may exist one, two or no fixed points of (9) with f de- fined as in (21). There is a slight subtlety, in that the nullclines may intercept where v > v_th, generating a vir- tual fixed point. From here on we refer to the branch of the v-nullcline with v < 0 (v > 0) as the left (right) v- branch. Since the system is PWL we may easily construct

the eigenvalues of fixed points, where they exist, as

2λ_± =

(1 − ω ±p(1 − ω)²− 4ω(β − 1), v > 0,

−s − ω ±p(s + ω)²− 4ω(β + s), v < 0. (22) Thus fixed points on the left v-branch are always stable, and the stability of fixed points on the right v-branch de- pends on the sign of 1 − ω. The exact nature of the fixed points is determined by the sign of the expression under the square root. Since β must be less than 1 to have two fixed points, the fixed point on the right v-branch is a saddle.

The sub-threshold dynamics are described by a contin- uous but non-differentiable system, so that the Jacobian matrix (around a fixed point) at the border separating linear subsystems is not defined. We shall call this bor- der the switching manifold, as crossing it causes a discon- tinuous change in the Jacobian. Nonsmooth bifurcations can occur as fixed points or limit cycles touch the switch- ing manifold under parameter variation. Importantly, the presence of a firing threshold in IF systems means that other nonsmooth bifurcations, and in particular grazing bifurcations as discussed in section 3, can arise.

The PWL-IF can generate periodic behaviour via a Hopf bifurcation (HB) of a fixed point on the right v- branch when ω = 1 (with β > 1) or through a discontin- uous Hopf-like (dHB, black line in Fig. 8) bifurcation at I = 0 (with ω < 1). We describe the second of these as be- ing discontinuous since the Jacobian around the fixed point changes discontinuously. The emergent sub-threshold limit cycle crosses through the switching manifold v = 0. Inter- estingly, with variation in I the frequency of the limit cycle does not change (and see [8] for a proof of this), whilst the amplitude grows linearly with I. As the limit cycle grows it can tangentially touch the firing threshold, causing a grazing bifurcation, whereupon sub-threshold oscillations are replaced by regular spiking solutions. In Fig. 8 we may observe both the dHB (black) and the grazing bifurcation (blue) in (I, β) parameter space.

Bistability can arise between a stable fixed point on the left v-branch and a fast spiking orbit when β > 1 and I < 0. In this parameter regime, there exists a saddle node on the right v-branch, which is key in delineating the basins of attraction of the two attractors. The basin of attraction of the stable fixed point is the union of the set of initial data such that trajectories reach threshold and are subsequently reset to the right of the separatrix of the saddle on the right v-branch. A homoclinic bifurcation (HC), indicated by the red curve in Fig. 8, will occur when the spiking limit cycle collides with the saddle, resulting in a homoclinic orbit from the saddle at the bifurcation point. Another form of bistability is also possible in this parameter regime, namely when a regular spiking limit cy- cle encloses the stable fixed point. The basin of attraction of this limit cycle is the set of points such that trajectories reach threshold and are reset to the right of the separatrix of the saddle (which is also enclosed by the stable spiking

orbit). Numerical studies suggest that the regular spiking orbit is lost as the basin of attraction of the stable fixed point grows and touches the orbit, and as such we shall call this an orbit crisis. As with the HC bifurcation, af- ter this point all trajectories will tend towards the stable fixed point. A plot of the basins of attraction of the two attractors is shown in Fig. 9, whilst a plot of parameter values for which we have an orbit crisis is depicted by the green (OC) curve in Fig. 8. For β < 1, we have a dis-

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

-6 -4 -2 0 2 4 6 8 10

I dHB

dSN HC

OC

SP GB

DB Quiescence

Subthreshold oscillations

Fast spiking

Regular spiking

D C

A B

`

OB

Figure 8: Bifurcation curves showing where solution types exchange stability in the (I, β) parameter plane. Other parameters are ω = 0.9, s = 0.35, vth = 60, vR = 20 and k = 0.4. The dHB refers to the discontinuous Hopf bifurcation, dSN refers to the discontinu- ous saddle node bifurcation, GB is the grazing bifurcation between sub-threshold oscillations and regular spiking ones, SP is the bifur- cation between the regular and fast spiking solutions, HC is the ho- moclinic bifurcation occurring when the fast spiking orbit collides with the saddle node, OC is the orbit crisis, marking the loss of the regular spiking solution, OB is the bifurcation marking the on- set/termination of bistability between sub-threshold oscillations and spiking ones, DB is the bifurcation marking the end of doublet fir- ing, the onset of which occurs along the SP curve. Regions A,B,C,D correspond to bistable parameter regimes, the solutions of which are depicted in figure 10. Solution types in the other regimes are marked.

continuous saddle node bifurcation (dSN, orange line in Fig. 8) at I = 0 where the saddle and stable fixed point come together and annihilate one another. We refer to this as a discontinuous bifurcation owing to the fact that the Jacobian of the system is undefined at the bifurcation point. For I > 0 there are no fixed points, and the only attractor is either the regular spiking or fast spiking orbit, dependent on the value of β. If β > 1 then the system only possesses one fixed point, which may be on the left or right v-branch dependent on the sign of I. As I crosses 0 from below, there are three scenarios: either ω > 1, in which case no change of stability occurs and trajectories tend to the fixed point, else ω < 1 and the fixed point becomes un- stable. We either may observe sub-threshold oscillations or spiking oscillations (either bursting or tonic) depending on the other parameter values. Using results from [47] we can say more about the sub- or super-critical nature of these bifurcations, though we do not pursue this here. As

β decreases through β_c = (v_th− I)/vth the fixed point no longer exists and we see spiking solutions only.

v a

0 10 20 30 40 50

0 10 20 30 40

Figure 9: Basins of attraction for the stable fixed point and limit cycle for ω = 0.9, β = 0.8, I = −0.2, s = 0.35, vth = 60, vR= 20 and k = 0.02. Black denotes the basin of attraction of the stable fixed point whereas white denotes the basin of attraction of the limit cycle.

We see that both basins are the union of disconnected sets. The green and yellow circles depict, respectively, the stable fixed point and saddle node whilst the purple dashed lines are the separatrices of the saddle node, given by the eigenvectors of the Jacobian there.

The separatrix separates the basin of attraction of the two attractors.

The large amplitude limit cycle is lost at the point where it is touches the basin of attraction of the stable fixed point.

-20 0 20 40 60

-20 0 20 40 60

v

a A

0 20 40

0 20 40 60

v

a B

0 20 40

0 20 40 60

v

a D

-20 0 20 40 60

-20 0 20 40 60

v

a C

Figure 10: Solution types in the regions indicated in Fig. 8. The blue and red solid curves indicate the periodic solutions; all solutions are stable. The orange dashed lines are the branches of the v-nullcline, whilst the skyblue dashed dashed depicts the a-nullcline. The green circles in the lower two figures are stable fixed points.

In parameter regimes where bursting orbits are stable, spikes are added when the a value after reset of the last spike of a bursting orbit crosses some value ac, resulting in a grazing bifurcation. The graze either occurs at v = 0, when the fixed point of (9), with f as in (21) is to the right of vR, or at v = vth if the fixed point is to the left of vR. After this point, trajectories will be forced up to thresh- old, so that the orbit gains an additional spike. For the case where the graze occurs at v = 0, the value of ac may 8

be found by integrating backwards from (v, a) = (0, I), the point at which (v, ˙v) = (0, 0), a time T , such that v(−T ) = v_R. The value of a_cis then equal to a(−T ). T is the flight time (in backwards time) from v = 0 to v = v_R and may be found numerically. For the case where the graze occurs at v = vth, the same method can be used, this time integrating from (v, a) = (vth, vth+ I). Inter- estingly, for bursting orbits, the value of ac may also be found by finding the curves of inflection of the vector field.

These curves separate trajectories that ‘bend’ rightwards, up to threshold, and those which ‘bend’ leftwards, towards the switching manifold, and are given by the solution to the equation d²a/dv² = 0. Substituting v = vR in the resulting equation will give ac For more discussion about inflection curves, we refer the reader to [8]. In the singular- limit as ω → 0, the inflection curve for v > 0 is precisely the right v-branch.

In Fig. 8, we concern ourselves only with non-bursting trajectories. In this case, the graze at v = v_thresults either in the transition from sub-threshold oscillations to spiking ones, or in the transition from regular spiking orbits to a 2-spike burst. The blue (GB) curve in Fig. 8 illustrates the first of these cases in (I, β) parameter space. Where v = 0, a graze results in the transition from fast to regular spiking, which may occur after a window of doublet firing.

The black curve (SP), in Fig. 8 corresponds to the tran- sition to regular spiking, either from fast spiking, or from doublet firing, whereas the pink curve (DB) marks the on- set/termination of doublet firing. We note that in order to have a graze at v_th we require that β > β_c since we need the v-nullcline to be below the a-nullcline for ˙v = 0 in this part of the phase-plane.

The number of spikes in a burst is controlled by varying either ω, I, v_R or v_th. Decreasing any of these parameters will result in bifurcations which decrease the number of spikes in a burst. Where vR < 0, the system is unable to burst as trajectories are always reset to the left of the right v-branch and are attracted to the left v-branch. We also note that we observe bursts for larger values of ω in the case where β > βc than where β < βc, and that large values of I may prohibit bursting, and we observe only fast spiking, so that I and β may be used as control parameters to switch between fast spiking and burst modes.

Owing to the discontinuous nature of the flow at reset, we may observe spiking orbits that enclose all other stable attractors, be they fixed points or sub-threshold oscilla- tions. The emergence of such orbits is controlled by the parameter k. Where k is too small, trajectories will sim- ply tend towards the attractors whose basin of attraction they are in. However, when k is large enough, we see the emergence of large amplitude limit cycles. These occur as the flows get ‘interrupted’ as they head towards an attrac- tor in the sub-threshold system. All trajectories starting outside these limit cycles are in the basin of attraction of such orbits.

We illustrate in Fig. 10 the stable solutions in the var- ious regions of parameter space indicated in Fig. 8. The

curves in Fig. 8 are generated by numerical continuation of solutions obtained from the firing map discussed later in section 6.

5. Periodic orbits and phase response curves To solve the PWL-IF model it is useful to recast the dynamics in matrix form so that:

X =˙

(A₁X + µ X₁≥ 0,

A2X + µ X1< 0, (23) where

A₁= 1 −1

ωβ −ω



, A₂=−s −1

ωβ −ω



, µ =I 0

 , (24) with X_i referring to the ith component of X (i.e. X₁= v and X₂= a). The solution to the equation ˙X = M X + µ can be written using matrix exponentials in the form

X(t) = G(t)X(0) + K(t)µ, (25) where

G(t) = e^{M t}, K(t) = Z t

0

G(s)ds. (26) Explicit solutions for G and K are easily constructed (and see for example [7]). Hereafter, we refer to Gⁱ, Kⁱ as the above expressions with the respective matrix M = Ai. To find a fast spiking orbit of period ∆ (in response to constant forcing) we need only solve (X1(∆), X2(∆)) = (vth, a0− k) subject to (X1(0), X2(0)) = (vR, a0), which gives a pair of simultaneous equations for (∆, a0) as:

vth= G¹₁₁(∆)vR+ G¹₁₂(∆)a0+ K₁₁¹ (∆)I, (27) a₀= G¹₂₁(∆)v_R+ K₂₁¹(∆)I + k

1 − G¹₂₂(∆) . (28)

Bursting orbits may be constructed using similar ideas, though with more book-keeping to keep track of the sub- trajectories (each determined by a linear system) that build the full periodic orbit. For example, for an orbit with

‘times-of-flight’ T_i^∗, i = 1, . . . N , (defined by the time spent in a region of phase space before meeting v = 0 or v = v_th) describing a bursting orbit with N − 2 spikes then we have to solve for the unknowns (T₁^∗, . . . , T_N^∗, a0) using a system of equations of the form

 0 a1



= G¹(T₁^∗)vR

a0



+ K¹(T₁^∗)µ,

 0 a2



= G²(T₂^∗) 0 a1



+ K²(T₂^∗)µ,

v_th a3



= G¹(T₃^∗) 0 a2



+ K¹(T₃^∗)µ, ...

v_th a_n



= G¹(T_n^∗)

 v_R a_n−1+ k



+ K¹(T_n^∗)µ, (29)

for n = 4, . . . , N subject to a₀ = a_N + k. The period of the orbit is simply ∆ =PN

i=1T_i^∗.

It is common practice in neuroscience to then charac- terize a neuronal oscillator in terms of its phase response to a perturbation. This gives rise to the notion of a phase response curve (PRC). The PRC quantifies the phase shift of an oscillator due to a small, brief perturbation as a func- tion of the phase of the oscillator when the perturbation occurred. A positive phase response indicates an advance- ment in the timing of the next oscillation, while negative values indicate a delay. For a detailed discussion of PRCs we refer the reader to [48]. One way to compute them for a given smooth dynamical system is via the Malkin adjoint method. Following [49] we briefly review this approach.

Consider a smooth dynamical system ˙z = F (z), z ∈ Rⁿ, with a ∆-periodic solution Z(t) = Z(t + ∆) and introduce an infinitesimal perturbation ∆z₀ to the trajectory Z(t) at time t = 0. This perturbation evolves according to the linearized equation of motion:

d∆z

dt = DF (Z(t))∆z, ∆z(0) = ∆z0. (30) Here DF (Z) denotes the Jacobian of F evaluated along Z.

Introducing a time-independent phase shift ∆θ as θ(Z(t)+

∆z(t)) − θ(Z(t)), we have to first order in ∆z that

∆θ = hQ(t), ∆z(t)i, (31)

where h·, ·i defines the standard inner product, and Q =

∇_Zθ is the gradient of θ evaluated at Z(t). Taking the time-derivative of (31) gives

 dQ dt, ∆z

= −

 Q,d∆z

dt

= −DF^T(Z)Q, ∆z . (32) Since the above equation must hold for arbitrary pertur- bations, we see that the gradient Q = ∇Zθ satisfies the linear equation

dQ

dt = −DF^T(Z(t))Q, (33)

subject to the conditions Q^T(0)F (z(0)) = 1/∆ and Q(t) = Q(t + ∆). The first condition simply guarantees that ˙θ = 1/T (at any point on the periodic orbit), and the second enforces continuity (and periodicity). The (vector) PRC, R, is related to Q according to the simple scaling R = Q∆.

In general (33) must be solved numerically to obtain the PRC, say, using the adjoint routine in XPP [50]. How- ever, for PWL models DF (Z) is piecewise constant, and we can obtain a solution in closed form [7]. Moreover it is also possible to extend the Malkin method to treat an IF process [32], which would give rise to a discontinuous PRC (at the spike time). In this latter case the continuity con- dition is swapped in favour of enforcing the normalisation condition Q^T(t)F (z(t)) = 1/∆ for all t.

For the PWL-IF model we may construct Q in given regions of phase space according to the prescription Q(t) =

J (T_i^∗− t)Q(T_i^∗), where J = G^T (and see [7] for further de- tails). Enforcing the normalisation condition at the times T_i^∗is enough to define a periodic (yet discontinuous) form for Q. For example, for a simple tonic spiking orbit we see that solving (33) and imposing the normalisation con- dition at t = 0 and t = ∆ gives a system of two linear equations in (q1, q2), where qiare the components of Q as

q1(∆)(vth+ I − a0+ k) + q2(∆)(ω(βvth− a0+ k) = 1

∆, q1(0)(vr+ I − a0) + q2(0)(ω(βvr− a0) = 1

∆. (34)

Using the further result that Q(0) = ΓQ(T ) where Γ = J¹(∆) for fast spiking orbits or Γ = J¹(T₃^∗)J²(T₂^∗)J¹(T₁^∗) for regular spiking orbits, gives

q2(∆) = r₁− r2Γ₁₁− r4Γ₂₁

T (r1(Γ12r2+ r4Γ22) − (r3r2Γ11+ r3r4Γ21)), q₁(∆) = 1

r₁

 1

∆ − r₃q₂(∆)



, (35)

where

r₁= v_th+ I − a₀+ k, r₂= v_R+ I − a₀, (36) r3= ω(vth− a0+ k), r4= ω(vR− a0). (37) Hence for a fast spiking orbit the adjoint is given by Q(t) = J (∆ − t)Q(∆) and for a regular spiking orbit the corre- sponding Q is

Q(t) =







J¹(T₁^∗− t)J²(T₂^∗)J¹(T₃^∗)Q(∆) 0 ≤ t ≤ t1

J²(T₂^∗− t)J¹(T₃^∗)Q(∆) t1≤ t ≤ t2

J¹(T₃^∗− t)Q(∆) t2≤ t ≤ ∆ ,

(38) where tj =Pj

i=1T_i^∗. In both cases the form of Q(∆) is given by (35). PRCs for bursting solutions may be con- structed in the same way, except that discontinuities are now not isolated to the ends of the periodic orbit, and so we must enforce both the normalisation condition just before and just after each threshold crossing. Typically, when studying neural oscillators, we are primarily con- cerned with the first (voltage) component of Q, since per- turbations to the system are usually given by changes in the external current, which acts only on the voltage vari- able. As an example we plot in Fig. 11 the voltage compo- nent of Q for a regular spiking orbit and a burst contain- ing three spikes. Knowledge of the PRC is fundamental in building network descriptions of weakly coupled oscilla- tors [51], and for PWL models is discussed in more detail in [7].

To study the stability of tonic spiking orbits (and for simplicity we focus here on the case that v > 0), we rewrite equation (23) as

dX

dt = M X + µ + dX

n

δ (t − T_n) , t ≥ 0 , (39) 10

-20 -10 0 10 20 30 40 50 60

0 1 2 3 4 5 6 7 8-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

t v

-20 -10 0 10 20 30 40 50 60

0 2 4 6 8 10 12 14 -1.5 -1 -0.5 0 0.5 1 1.5 2

t

PRC

Figure 11: Left: Voltage component of the phase response curve for a regular spiking orbit (red, solid). Right: Voltage component of the phase response curve for a 3-spike bursting orbits (red, solid).

Parameter values are β = 1.1, s = 0.35, k = 0.4 and ω = 1 for the regular spiking orbit and ω = 0.25 for the bursting orbit. Corre- sponding orbits are shown in dashed blue.

with

d =v_R− vth

k



. (40)

Integrating equation (39) between two successive firing times yields the closed form expression

X⁻(Tn+1) = G(∆n)[X⁻(Tn) + d] + K(∆n)µ, (41) with ∆_n = T_n+1− T_n. The superscript on X in equa- tion (41) indicates that we evaluate X at the firing event before the reset, i.e. X⁻(T_n) = lim_ε&0X (T_n− ε). For later reference, we here also introduce

X⁺(Tn) = limε&0X (Tn+ ε) and note that X⁺(Tn) = d + X⁻(Tn). A perturbation of the periodic orbit s with a period ∆ leads to perturbed firing times eTn, for which we make the ansatz eTn = n∆ + δTn. Similarly, we write the perturbed trajectory as eX(t) = s(t) + δX. Hence, we have from equation (41)

Xe⁻(T_n+1) = G( e∆_n)[ eX⁻( eT_n) + d] + K( e∆_n)µ, (42) where e∆n = eTn+1− eTn. Linearising equation (42) then results in

δXn+1= e^{M ∆}δXn− δTne^{M ∆}p + δTn+1q , (43) with

p = Ms⁻(∆
+ d] + µ = ˙s⁺(∆) , (44a) q = M s⁻(∆) + µ = ˙s⁻(∆) , (44b) where δX_n is defined through eX(T_n) = s(T_n) + δX_n and

˙s = ds/dt. At first sight, equation (43) appears to be implicit since δX_n+1 is given in terms of the unknown perturbation of the firing time δTn+1. However, we need to solve equation (43) with the constraint that ev( eT_n⁻) =

v_th = v(T_n⁻), so that the first component of δX_n+1 van- ishes. Defining the row vector γ with components γ_i =

−[e^{M ∆}]_1i/[q]₁, we find for the perturbed firing time δTn+1= γ (δXn− pδTn) , (45) which immediately leads to

δXn+1= e^{M ∆}+ qγ
(δXn− pδTn) . (46) Hence, the perturbations of X at the (n + 1)th firing time are uniquely determined by the perturbations at the nth firing event. From equation (45), we see that

δXn− pδTn= e^{M ∆}− M dγ
(δXn−1− pδTn−1) . (47) Without loss of generality, we set δT₀= 0 at t = 0, which is equivalent to saying that there is some perturbation of the periodic orbit at t = 0. Then, equations (45) and (46) yield δT₁= γδX₀and δX₁= e^{M ∆}+ qγ
δX0, so that we find from equations (46) and (47)

δXn+1= e^{M ∆}+ qγ

e^{M ∆}− M dγ
ⁿ

δX0. (48) Hence, the perturbations grow without bound if there is at least one eigenvalue of B = e^{M ∆}− M dγ
with mod- ulus larger than one. Conversely, if all eigenvalues of B have moduli smaller than one, then any initial perturba- tion decays towards zero. However, our analysis indicates that B always possesses exactly one eigenvalue equal to 1, so that Bⁿ converges against a constant matrix B for large n instead of decaying if all other eigenvalues have moduli smaller than one. The stability of the period one orbit is then determined by the product e^{M ∆}+ qγ
B.

For the parameter values where fast spiking orbits exist (e.g. I = 4.0, k = 0.4, vth = 60, vR = 8.1, β = 0.5, s = 0.35, ω = 0.08) we find that this product equals zero, so that the orbit is asymptotically stable. The above ar- gument relies on the numerical evaluation of the matrices and eigenvalues. A more detailed study on the structure of B will be reported elsewhere.

Next we show how to determine single neuron behaviour (existence and stability) via an alternative approach based on the construction of a discontinuous one-dimensional re- turn map.

6. Firing map

Due to the nature of the nonsmooth dynamics of the system at reset, it is useful to consider a map of the adap- tation variable at successive firing times. This will collapse the dynamics of the full system to a one-dimensional return map. This has previously been considered by Touboul and Brette [52] for a broad class of planar nonlinear IF models.

Here, we focus on the construction of such a map for the PWL-IF model. We consider a set, called the Poincar´e section, Σ = {(v, a)|v = ¯v ∈ R} which is transverse to the flow for all (¯v, a) ∈ Σ. The value of ¯v above is arbitrary,